Rational Equations

A rational equation is an equation that can be written as

 Where N(x) and D(x) are polynomials.

 

 The Domain of a rational function includes all real numbers except for any values where D(x)=0.  this is because if D(x)=0, The function would be undefined.  However, the zeros of D(x) are still important to rational functions.  At the zeros of D(x), there is an imaginary line known as a vertical asymptote.  As the Graph approaches the asymptote, the Y values will go to infinity and negative infinity but never cross the line where D(x)=0.  

Graphs can also have horizontal asymptotes.   

If the degree of N(x) is Greater than the degree of D(x), then there is no horizontal asymptote.

If the degree of N(x) is less than the degree of D(x), then there is a horizontal asymptote at Y=0.

If the degree of N(x) is equal to the degree of D(x),then the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and denominator.

Other points of interest include:

X-intercepts: when f(x)=0 or more simply, when N(x)=0.

Y-intercept: f(0)

Example:

Based on the equation, we can determine that there will be a vertical asymptote at x=3 and a horizontal asymptote at y=2.
By solving for f(0), we can determine that the y-intercept will be at (0, 0)
By solving for f(x)=0, we can determine that there is an x-intercept at (0, 0)

In order to describe what the graph is doing at the vertical asymptote, we must use limit notation.

Saying that the graph decreases as it moves toward the Asymptote from the left, we say

The Superscript - on the 3 indicates the left side of the graph.
Thus in order to describe the Right side of the graph we use a superscript + on the 3.

Polynomial Functions of Higher Degree

0 Degree

1st Degree

3rd Degree

2nd Degree

4th Degree

5th Degree

Above are the graphs of varying polynomial functions. As you can see, as the degree of the function increases, the graph gets curvier. While this is not always the case, it is a general rule of thumb.

End Behavior

The ends of the graphs of the varying polynomial functions are actually pretty predictable. By looking at the leading coefficient of the term with the highest degree and the degree of the function, one can figure out how the function will look when graphed.

A positive leading coefficient will make the right side of the right most extreme on the graph ALWAYS approach infinity. This is written as such in limit notation:

A negative leading coefficient will make the right side of the right most extreme on the graph ALWAYS approach negative infinity. This is written as such in limit notation:

The opposite end of the graph either goes the same way as the right side or does the opposite. If the degree is EVEN, the left side acts the SAME as the right. If it is ODD, the left side acts the OPPOSITE of the right side.

Zeroes

A polynomial function has as many zeroes as its degree; however, it does not necessarily have that many X-Intercepts. A polynomial may have imaginary zeroes which are not graphed on the Cartesian plane.

Sometimes roots are repeated and can make a function that has two x intercepts only have one. This is called multiplicity.

This graph should have 4 X-Intercepts, however it only has 3. At X=0 the curve runs tangent to the X-Axis

Extrema

The graph of a polynomial function may have as many relative maxima/minima as (n-1) where n is the degree of the polynomial

Review

The graphs of polynomial functions have certain rules that allow you to predict what they will look like pretty easily. Knowing the degree and the leading coefficient allows you to make a sketch of the graph that will resemble the actual graph.